8种工程设计优化问题Matlab代码：多种智能优化算法求解对比

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一、8种工程设计优化问题

二、8种工程设计优化问题部分问题介绍

2.1 焊接梁设计

2.2 拉力压缩弹簧设计

2.3 三杆衍架设计问题

2.4 压力容器设计

2.5 减速器设计问题

三、多种智能优化算法分别求解部分工程设计优化问题

3.1 焊接梁设计

3.2 拉力压缩弹簧设计

3.3 三杆衍架设计问题

四、8种工程设计优化问题Matlab代码

一、8种工程设计优化问题

八种工程设计问题有：Welded beam design (焊接梁设计)、Four-stage gear box problem (四级变速箱问题)、Tension compression spring design (拉力压缩弹簧设计）、Three-bar truss design problem (三杆衍架设计问题)、Weight Minimization of a Speed Reducer (减速器重量最小化)、Rolling element bearing (滚动轴承)、Pressure vessel design (压力容器设计)、Gear train design Problem (齿轮系设计问题)。

二、8种工程设计优化问题部分问题介绍

2.1 焊接梁设计

焊接梁设计问题 (Welded Beam Design, WBD) 是一个最小化问题，其中优化算法是为了降低设计的制造成本。该优化问题可以描述为寻找满足切应力 ( $\tau$ )、弯曲应力 (θ)、梁条弯曲载荷( $P_{c}$ )，末端偏差( $\delta$ )和边界条件等约束的四个设计变量——即梁条的长度( $l$ )、高度( $t$ )、厚度( $b$ )和焊缝厚度( $h$ )，使得制造焊接梁的费用最小，因此焊接梁问题是一个典型的非线性规划问题。

2.2 拉力压缩弹簧设计

拉压弹簧设计问题的目的是在满足最小挠度、震动频率和剪应力的约束下，最小化拉压弹簧的重量。该问题由３个连续的决策变量组成，即弹簧线圈直径（ $d$ 或 $x_{1}$ )、弹簧簧圈直径（ $D$ 或 $x_{2}$ ) 和绕线圈数（ $P$ 或 $x_{3}$ )。数学模型表示公式如下：

2.3 三杆衍架设计问题

三杆桁架设计问题的目的是通过调整横截面积（ $x_{1},x_{2}$ ）来最小化三杆桁架的体积。该三杆式桁架在每个桁架构件上受到应力（σ ）约束，该优化问题具有一个非线性适应度函数、３个非线性不等式约束和两个连续决策变量，如下所示：

2.4 压力容器设计

压力容器设计问题的目标是使压力容器制作(配对、成型和焊接)成本最小，压力容器的设计如下图所示压力容器的两端都有盖子封顶，头部一端的封盖为半球状。 $L$ 是不考虑头部的圆柱体部分的截面长度， $R$ 是圆柱体部分的内壁直径， $T_{s}$ 和 $T_{h}$ 分别表示圆柱体部分壁厚和头部的壁厚， $L$ 、 $R$ 、 $T_{S}$ 和 $T_{h}$ 即为压力容器设计问题的四个优化变量。

2.5 减速器设计问题

减速器设计问题是一个工程设计问题。它有7个设计变量，如下图所示。本设计问题的主要目标是在满足以下约束条件的情况下使减速器的重量最小化：齿轮齿的弯曲应力、表面压力、轴的横向偏转、轴上的应力。有7个设计变量(z1-z7)，如面宽(b)，齿的模块(m)，齿轮齿数(p)，轴承之间的第一轴的长度( $l_{1}$ )，轴承之间的第二轴的长度( $l_{2}$ )，第一轴的直径( $d_{1}$ )，第二轴的直径( $d_{2}$ )。

三、多种智能优化算法分别求解部分工程设计优化问题

3.1 焊接梁设计

The best solution obtained by BWO is : 0.125 7.4444 6.8811 0.39371
The best optimal values of the objective funciton found by BWO is : 2.9235
The best solution obtained by AO is : 0.34224 2.9617 5.0921 0.66614
The best optimal values of the objective funciton found by AO is : 3.1512
The best solution obtained by GWO is : 0.19074 3.5256 9.1937 0.1995
The best optimal values of the objective funciton found by GWO is : 1.6881
The best solution obtained by AVOA is : 0.13451 5.3821 8.5675 0.22889
The best optimal values of the objective funciton found by AVOA is : 1.9361
The best solution obtained by WOA is : 0.43465 2.1312 5.2844 0.61559
The best optimal values of the objective funciton found by WOA is : 2.9694
The best solution obtained by GTO is : 0.27338 2.6244 7.8878 0.27339
The best optimal values of the objective funciton found by GTO is : 1.9414
The best solution obtained by MFO is : 0.16886 4.0143 9.1848 0.1992
The best optimal values of the objective funciton found by MFO is : 1.7121
The best solution obtained by SO is : 0.21292 3.1196 9.1816 0.22292
The best optimal values of the objective funciton found by SO is : 1.842

3.2 拉力压缩弹簧设计

The best solution obtained by BWO is : 0.05 0.317371 14.4544
The best optimal values of the objective funciton found by BWO is : 0.013055
The best solution obtained by AO is : 0.065405 0.71716 4.0043
The best optimal values of the objective funciton found by AO is : 0.01842
The best solution obtained by GWO is : 0.05 0.317169 14.079
The best optimal values of the objective funciton found by GWO is : 0.012749
The best solution obtained by AVOA is : 0.060251 0.59982 4.3835
The best optimal values of the objective funciton found by AVOA is : 0.0139
The best solution obtained by WOA is : 0.057822 0.52301 5.609
The best optimal values of the objective funciton found by WOA is : 0.013306
The best solution obtained by GTO is : 0.055032 0.44261 7.5935
The best optimal values of the objective funciton found by GTO is : 0.01286
The best solution obtained by MFO is : 0.068994 0.93343 2
The best optimal values of the objective funciton found by MFO is : 0.017773
The best solution obtained by SO is : 0.059675 0.58102 4.6413
The best optimal values of the objective funciton found by SO is : 0.013741

3.3 三杆衍架设计问题

The best solution obtained by BWO is : 0.85745 0.2773
The best optimal values of the objective funciton found by BWO is : 270.2535
The best solution obtained by AO is : 0.7375 0.58715
The best optimal values of the objective funciton found by AO is : 267.3117
The best solution obtained by GWO is : 0.77852 0.43803
The best optimal values of the objective funciton found by GWO is : 264.0021
The best solution obtained by AVOA is : 0.83312 0.29512
The best optimal values of the objective funciton found by AVOA is : 265.1538
The best solution obtained by WOA is : 0.74184 0.55994
The best optimal values of the objective funciton found by WOA is : 265.8196
The best solution obtained by GTO is : 0.78863 0.40839
The best optimal values of the objective funciton found by GTO is : 263.8958
The best solution obtained by MFO is : 0.79437 0.39237
The best optimal values of the objective funciton found by MFO is : 263.9192
The best solution obtained by SO is : 0.78232 0.42653
The best optimal values of the objective funciton found by SO is : 263.9275